Locally finite type and finite type morphisms

Depends on

• Morphisms of schemes
• Affine open subschemes
• Subalgebra generated by a subset, algebras of finite type, and module-finite algebras

Used by

• Euler characteristic in a proper flat family is locally constant Corollary
• Finite morphisms are proper Corollary
• General hypersurfaces give smooth complete intersections Corollary
• Generic smoothness on the source Corollary
• Global functions on geometrically connected and geometrically reduced proper schemes Corollary
• Proper birational normal curves agree off finitely many points Corollary
• Upper semicontinuity of fibre cohomology dimensions Corollary
• An infinite disjoint union is locally but not globally finite type Counterexample
• Finite type need not mean finite presentation Counterexample
• Unramified of finite presentation does not imply flat or etale Counterexample
• Algebraic cycles and the cycle group of a scheme of finite type over a field Definition
• Birational morphisms of integral finite-type schemes Definition
• Canonical bundle and canonical divisors Definition
• Canonical resolutions with invariants of a marked ideal Definition
• Curves over a field Definition
• Group schemes of finite type over a field Definition
• Neron models, the Neron mapping property and weak Neron models Definition
• Proper morphisms Definition
• Pushforward of coherent sheaves in algebraic K-theory Definition
• Quasi-finite morphisms of schemes Definition
• Rational maps of integral finite-type schemes Definition
• Scheme-theoretic varieties Definition
• Smoothness over a field by geometric regularity Definition
• The intrinsic Zariski tangent space Definition
• The Riemann-Roch dimension l(D) Definition
• Unramified morphism Definition
• A closed point immersion is unramified Example
• A nondegenerate projective quadric Example
• Affine n-space over an arbitrary base Example
• Divisor of a rational function on the projective line Example
• Principal divisors on the projective line have degree zero Example
• The empty morphism is finite, proper and projective Example
• The Hopf algebra of a split torus and its root-of-unity subgroups Example
• A transverse hyperplane slice is smooth at the chosen point Lemma
• Affine finite-type source immerses into relative projective space Lemma
• An affine scheme of finite type over a field has a finitely generated coordinate ring Lemma
• Birational morphisms restrict to isomorphisms between principal affine opens Lemma
• Chow lemma for proper Noetherian schemes Lemma
• Closed gluing of two projective three-spaces is proper Lemma
• Composite of a finite morphism and a proper morphism is proper Lemma

…and 53 more results.

Dependency tree · two levels

11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• The Stacks Project, Morphisms of Schemes, Definition 15.1 (standard reference, not scraped)